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Curran919

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Reply with quote  #16 
Oooookay, I can follow that better now. Yeah, I guess the mass and the deflection at tip is all inherent in the Force term. Looks good to me [thumb]
electricpete

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Reply with quote  #17 

Thanks Curran. I appreciate if when you review and comment on anything I post  because I make plenty of mistakes and you often bring up some great points. 

If you guys will excuse me wandering a little more, … I'd like to record some more thoughts about the general subject of stress in absorbers that arose from the previous discussion. 

Let's redo that massless beam model in an even simpler fashion: without the constraints of geometric scaling:

  • Sigma_max := h/2 * Moment_max/Ibending;
  • Where Moment_max = Fabs * L,  I bending = w*h^3/12
  • Sigma_max = (h/2) * (Fabs*L) / (w*h^3/12) = 6 * Fabs*L / (w*h^2)     (where Fabs is constant)

Now we see that in order to minimize stress we simply want to minimize the length L and maximize the cross section parameters w and h, especially h (thickness) which has an exponent of 2.  (M will have to adjust to maintain frequency, and of course also higher M deermines frequency separation)

We can see directly from the above simple equations where each of these dependencies comes from:  

  • Lower length L decreases max stress because it moves the applied force closer to the base, creating lower moment arm => lower moment from that same force.
  • Higher thickness h tremendously helps increase the bending moment of inertia which shows in the denominator of sigma_max, but reduced one exponent since the bending stress grows linearly  as we move from the center out along the thickness toward the edges (thicker bar has more distance for bending stress to grow between center and edge).  So the exponent is 3-1 = 2.  Doubling the thickness h reduces the max stress by 4 (everything else constant except for mass which is adjusted to keep same resonant frequency).
  • Higher width w also helps by virtue of appearing to the first power in the bending moment of inertia.

All the above for massless beam.  Distributed mass beam is a way more  complicated exercise which surely doesn't have identical results, but I'm thinking  this simple massless model is a good good starting point to think about it and each of these variables affects stress qualitatively in the same direction stated (shorter beam with larger cross section tends to reduce max stress).  Also I said before that I'd prefer to put as much weight as possible in the beam vs the attached mass.  That was based on brute force numerical trial and error, but I think we can connect it to the results above: Putting a lot of mass into the beam moves some of the mass-acceleration force down along the beam where it is closer to the base and creates less moment arm.  Maybe that's an inexact fuzzy way to think about it, but it's a starting point. 

Sorry Oli for pulling your thread off topic by misunderstanding your post and rambling about stress.  

OLi

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Reply with quote  #18 
No problem, you are welcome any time and question may pop up.
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Curran919

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Reply with quote  #19 

But you aren't properly considering how the geometry changes in order to hold the natural frequency constant:

Sigma_max = 6 * Fabs*L / (w*h^2)
I bending = w*h^3/12
k of a cantilever beam = 3*E*I/L^3 = E*(w*h^3)/(4*L^3) 
and fn is proportional to sqrt(k/m), so in order to keep the fn constant:

  • a halving of 'height' can be offset by halving length. This also doubles sigma_max.
  • a halving of 'height' can be offset by multiplying width by 8. This also halves sigma_max.
  • a doubling of length can be offset by multiplying width by 8. This also quarters sigma_max.
So clearly for a massless beam, in order to minimize max stress, you want a long length, and a very wide shaft, with the height/thickness not being a driving factor (use that as the tuning parameter).
electricpete

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Reply with quote  #20 

Quote:
But you aren't properly considering how the geometry changes in order to hold the natural frequency constant:

My stated assumption was "(M will have to adjust to maintain frequency, and of course also higher M determines frequency separation)".    So it was considered, in a way that lends itself to simplicity because can look at the effect of changing only one variable at a time among the dimensions that that show up in the stress equations (the compensating change in mass doesn't show up in the stress equations).  

You provided an alternative way to look at the same relationships, with assumption of constant mass.  Assuming the preliminary/rough design thought process starts with a target M for frequency separation, then your characterization is a more useful way to look at the relationships (to the extent the massless assumption is useful in rough design...maybe some mounting situations may dictate small cross section bar).  Both these different ways to look at it are useful imo.   Thanks for adding yours.... I'm still pondering what all this might tell us for the distributed mass scenario... 

...For example is the above simple calculation of max stress (for a given Fabs) from beam dimensions under massless assumption a bounding-high / worst-case estimate relative to the distributed mass beam absorber case with  the same beam dimensions? I think it is.  My simple way to think of it is that we integrate Fabs twice along the length of beam from end to base in order to find Moment at the base... the way to get the highest result is to stack all the Fabs (mass) at the end, which corresonds to the massless beam case.   Even though we don't know the mode shape of the distributed case (which affects the conversion between mass and force dF =  y[x] w^2 mu dx), it doesn't matter in this particular conclusion since we do know that the total Fabs remains the same and the worst case location to stack that fixed Fabs must be at the end of the beam.  So beam dimensions alone do provide a bounding/worst case figure of merit (stress per Fabs) for an absorber design.  I'm pretty sure it is true/correct. But even so, I guess it's not tremendously practical.   The value of knowing the worst case in quantitatively evaluating stress in a single absorber design is limited if we don't know this Fabs (which we don't).  The value of this figure of merit in qualitatively comparing different absorber designs is diluted because it's not a direct comparison but a comparison of worst cases. 

abrahamr

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Reply with quote  #21 
Quote:
Originally Posted by Edwin
The passive damper should work with any vibration as long as it is at its frequency. 
To determine if there is resonance, you have a few things you can check. First is to look at the width of the peak in the spectrum. A resonance will show a wide foot of it. It also will occur in one direction much stronger than in orthers.
If you have the possibility do stop and start the equipment (or just change speed enough) you can measure is there is a phase shift around this frequency. That will be there with resonance but not without it.



Hi, I am working with OLi in this project. Thanks for the feedback.

Below are actual measurements from site (first the spectrum and second a zoom-in). To me, the peak does not show any fat tails/wide foot. What do you think?

We can only vary the speed before the unit is synchronized to the grid, but at that point there is no load and hardly any vibrations (I don't think the peak is measurable). We can start, synchronize, load and stop the unit multiple times, so that would be testable. But the speed would be the same every time. Would that tell us anything?

spectra.png  zoom.png 
Walt Strong

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Reply with quote  #22 
"To me, the peak does not show any fat tails/wide foot. What do you think?"

I usually look at spectrum with Log-amplitude for possible natural frequency. I also use high spectrum resolution with a lot of averages (about 20+) and have found natural frequency peak separated from forcing frequency.

Have you verified frequency is not related to shaft rotation by time synchronous averaging? I would measure wicket vibration torsional (tangential to shaft) while changing gate position to see if amplitude changes from possible vortex flow off (discharge) of gates or from upstream flow disturbance. I prefer to understand cause of the unusual frequency (yes, low amplitude) before trying to install a mass damper. Nevertheless, since this is apparently an old problem that had a damper solution, then I suppose you could proceed with a new damper! Too many ways to look bad, if the new damper does not work well or if it masks a future structural fatigue failure!

Walt
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